Cordeau_DialARide

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DOI: 10.1007/s10288-002-0009-8

4OR 1: 89101 (2003) Invited Survey

The Dial-a-Ride Problem (DARP):Variants, modeling issues and algorithms

Jean-Franois Cordeau, Gilbert Laporte

GERAD-HEC Montral, 3000 chemin de la Cte-Sainte-Catherine,Montral, Canada H3T 2A7 (e-mail: [email protected])

Received: June 2002 / Revised version: August 2002

Abstract. The Dial-a-Ride Problem (DARP) consists of designing vehicle routesand schedules for n users who specify pick-up and drop-off requests between ori-gins and destinations. The aim is to plan a set of m minimum cost vehicle routescapable of accommodating as many users as possible, under a set of constraints. Themost common example arises in door-to-door transportation for elderly or disabledpeople. The purpose of this article is to review the scientific literature on the DARP.

The main features of the problem are described and classified and some modelingissues are discussed. A summary of the most important algorithms is provided.

Keywords: dial-a-ride problem, survey, static and dynamic pick-up and deliveryproblems

AMS classification: 90B06, 90C27, 90C59

1 Introduction

The Dial-a-Ride Problem (DARP) consists of designing vehicle routes and sched-ules for n users who specify pick-up and drop-off requests between origins anddestinations. Very often the same user will have two requests during the same day:an outbound request from home to a destination (e.g., a hospital), and an inboundrequest for the return trip. In the standard version, transport is supplied by a fleet ofm identical vehicles based at the same depot. The aim is to plan a set of minimumcost vehicle routes capable of accommodating as many requests as possible, under aset of constraints. The most common example arises in door-to-door transportationservices for elderly or disabled people (Madsen et al. 1995; Toth and Vigo 1996,1997; Borndrfer et al. 1997).

4ORQuarterly Journal of the Belgian, Frenchand Italian Operations Research Societies

Springer-Verlag 2003


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90 J.-F. Cordeau and G. Laporte

In western countries several local authorities are setting up dial-a-ride servicesor are overhauling existing systems in response to increasing demand. This phe-nomenon can be attributed in part to the ageing of the population but also to a trendtoward the development of ambulatory health care services. Some existing systems

cannot adequately meet demand while others are faced with escalating operatingcosts. There is a genuine need for reliable cost effective systems, and operationalresearch can help reach this goal.

From a modeling point of view, the DARP generalizes a number of vehicle rout-ing problems such as the Pick-up and Delivery Vehicle Routing Problem (PDVRP)and the Vehicle Routing Problem with Time Windows (VRPTW). For overviews onthese problems, see Desrosiers et al. (1995) and Desaulniers et al. (2002). Whatmakes the DARP different from most such routing problems is the human per-spective. When transporting passengers, reducing user inconvenience must be bal-

anced against minimizing operating costs. In addition, vehicle capacity is normallyconstraining in the DARP whereas it is often redundant in PDVRP applications,particularly those related to the collection and delivery of letters and small parcels.

The purpose of this article is to review the scientific literature specific to theDARP. It is organized as follows. In Sect. 2 the main features of the DARP aredescribed and classified, and some modeling issues are discussed. A summary ofthe most important algorithms is provided in Sect. 3, followed by conclusions inSect. 4.

2 Main features of the DARP

Dial-a-ride services may operate according to a static or to a dynamic mode. In thefirst case, all transportation requests are known beforehand while in the second caserequests are gradually revealed throughout the day and vehicle routes are adjustedin real-time to meet demand. In practice pure dynamic DARPs rarely exist since asubset of requests is often known in advance.

Most studies on the DARP assume the availability of a fleet ofm homogeneousvehicles based at a single depot. While this hypothesis often reflects reality andcan serve as a sound base for the design of models and algorithms, it is importantto realize that different situations exist in practice. There may be several depots,especially in wide geographical areas, and the fleet is sometimes heterogeneous.Some vehicles are designed to carry wheelchairs only, others may only cater toambulatory passengers and some are capable of accommodating both types ofpassenger. The main consideration in some problems is to first determine a fleetsize and composition capable of satisfying all demand while in other contexts, theaim is to maximize the number of requests that can be served with a fixed sizefleet. Some systems routinely turn down several requests each day. A compromiseconsists of serving some of the demand with a core vehicle fleet and using extravehicles (e.g., taxis) if necessary.


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The Dial-a-Ride Problem 91

Given this, it makes sense to consider two possible problems: 1) minimizecosts subject to full demand satisfaction and side constraints; 2) maximize satisfieddemand subject to vehicle availability and side constraints. The most common costelements relate to regular fleet size and operation, occasional use of extra vehicles,

and driver wages.Quality of service criteria include route duration, route length, customer waiting

time, customer ride time (i.e., total time spent in vehicles), and difference betweenactual and desired drop-off times. Some of these criteria may be treated as con-straints or as part of the objective function. A common trend in DARP models isto let users impose a time window on both their departure and arrival times. Webelieve this may be unduly constraining for the transporter, particularly if thesetime windows are narrow. Following Jaw et al. (1986) we believe that users shouldbe able to specify a time window on the arrival time of their outbound trip and on

the departure time of their inbound trip. The transporter then determines a planneddeparture time for the outbound trip and a planned arrival time for the inbound trip,while satisfying an upper bound on the ride time. In practice, since travel times aresomewhat uncertain, the outbound departure time communicated to the user shouldbe slightly earlier than the scheduled time.

3 Algorithms

Three important decisions are associated with the construction of a DARP solution:1) determining clusters of users to be served by the same vehicle; 2) sequencingtheseusersintoavehicleroute;3)scheduling pick-up,drivinganddrop-offactivitiesalong each route. Some algorithms execute these steps sequentially while otherstake a more holistic perspective and intertwine these decisions. We will first presentthe scheduling aspect which plays an important role in several DARP algorithms.This will be followed by a description of the best known algorithms under two

headings: single-vehicle DARP and multi-vehicle DARP.

3.1 Scheduling

Given a route k = (v0, . . . , vi , . . . , vq ) consisting of a sequence of vertices, wherev0 and vq both represent the depot, the scheduling problem is to determine thedeparture time from the depot and the time at which service should begin at eachvertex v1, . . . , vq1, so that time windows are satisfied and route duration is mini-mized. This problem is of critical importance whenever an upper bound is imposedon route duration.


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We use the following notation:

Tk : the maximal duration of route k;[ei , i] : a time window on the beginning of service at vertex vi (every vehicle

must leave the depot no earlier than e0 and return no later than 0);tij : the travel time from vi to vj ;di : the service duration at vi ;Ai : the arrival time of the vehicle at vi ;Bi : the time at which service begins at vi ;Di : the departure time from vi ;Wi : the waiting time at vi .

Note that Bi max{ei , Ai} and Di = Bi + di . The time window at vi is

violated ifBi > i . Arrival at vi before ei is allowed and therefore the waiting timeat that vertex is Wi = Bi Ai .If the scheduling problem is feasible, a solution can be identified by sequentially

setting B0 = e0 and Bi = max{ei , Ai} for i = 1, . . . , q. To reduce route durationand unnecessary waiting time, it may be advantageous to delay departure from thedepot and the beginning of service at pick-up vertices. For this, one must computefor each vi , the maximum delay Fi that can be incurred before service starts so thatno time window in route k will be violated. Savelsbergh (1992) calls Fi theforwardtime slackofvi . It is computed as

Fi = minijq

j

Bi +

ip


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the successor ofvi in route k, the problem can be formulated more generally as anoptimization problem of the form

Minimize

q

i=0

gi (Bi ) (3)

subject toBi Bi+1 ti,i+1 di (i = 0, . . . , q 1) (4)

Bi ei (i = 0, . . . , q ) (5)

Bi i (i = 0, . . . , q ) , (6)

where gi (Bi ) is a convex function defined with respect to the time window [ei , li].Dumas et al. (1989) have proposed a dual approach to solve this problem by per-

forming q unidimensional minimizations. In the special cases where the inconve-nience functions are quadratic or linear, the complexity of the algorithm is O(q).

Finally, Hunsacker and Savelsbergh (2002) have devised a procedure for effi-ciently testing the feasibility of an insertion in construction or improvement heuris-tics. They consider a variant of the DARP with time windows, an upper bound onWi , and an upper bound on the ride time, proportional to the driving time. Theyhave shown how to check in O(q) time whether the insertion of a given request ina route is feasible.

3.2 The single-vehicle DARP

One of the simplest cases of the DARP is where all requests are known in advanceand all users are served by a single-vehicle. Psaraftis (1980) formulated and solvedthe problem as a dynamic program in which the objective function is the minimiza-tion of the weighted sum of route completion time and customer dissatisfaction.Customer dissatisfaction is itself expressed as a weighted combination of waitingtime before pick-up and ride time. Time windows are not specified by users. In-stead the transporter imposes maximum position shift constraints limiting thedifference between the position of a user in the calling list and its position in thevehicle route. This algorithm was later updated by the same author (Psaraftis, 1983)to handle user-specified time windows on departure and arrival times. As is oftenthe case in dynamic programming formulations, the algorithm can only solve rel-atively small instances optimally since the procedure has an O(n23n) complexity.The largest instance solved using this approach contains nine users. While mostDARPs arising in practice are much larger, the proposed approach could still proveuseful as a subroutine in a multi-vehicle algorithm, provided the number of usersin each route remains relatively small.

Sexton (1979) and Sexton and Bodin (1985a, 1985b) also view the single-vehicle DARP as a step in a multi-vehicle DARP heuristic in which the usershave previously been clustered. Their algorithm iterates between solving a routing


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problem by means of an insertion heuristic and solving the associated schedulingproblem. They formally describe the alternation of these two steps in the contextof Benders decomposition. These authors minimize a user inconvenience functionmade up of the weighted sum of two terms. The first measures the difference

between the actual travel time and the direct travel time of a user. The second termis the (positive) difference between desired drop-off time and actual drop-off time,under the assumption that the former is at least as large as the latter, late drop-offsbeing disallowed. As explained in Sect. 3.1, this objective can be expressed as alinear function of the Bi variables. Results are reported on several data sets fromBaltimore and Gaithersburgh, where the number of users varies between 7 and 20.

The single-vehicle DARP was reformulated as an integer program by Desrosierset al. (1986). The formulation includes time windows as well as vehicle capacityand precedence constraints and it is solved exactly by dynamic programming. Us-

ing a double labelling scheme, the authors were capable of identifying and latereliminating several dominated states and state transitions. Optimal solutions wereobtained for n = 40.

The dynamic single-vehicle DARP was also considered by Psaraftis (1980). Inthis problem, new requests occur dynamically in time but no information on futurerequests is available (unlike what happens in stochastic programming). When anew request becomes known at time t a planned solution is available. All requestsscheduled before t have already been processed and are no longer relevant. Theproblem is then to reoptimize the portion of the solution from time t, including

the new request. This is done by applying the dynamic programming algorithmdeveloped for the static case. One practical difficulty stemming from this approachis being capable of solving the problem at time t before the arrival of the nextrequest, which may not be feasible if the algorithm is slow and requests arrive inquick succession. One way around this difficulty, recently proposed by Gendreau etal. (2001) in the context of dynamic ambulance relocation, is to precompute severalscenarios, using parallel computing, in anticipation of future requests. Despite itslimitations,Psaraftiss work on the dynamic single-vehicle DARP has helped definethe concepts used in later research on dynamic routing problem (see Psaraftis, 1988;

Psaraftis 1995; Mitrovic-Minic et al. 2002).

3.3 The multi-vehicle DARP

One of the first heuristics for the multiple-vehicle static DARP was proposed byJaw et al. (1986). The model considered by these authors imposes windows on thepick-up times of inbound requests and on the drop-off times of outbound requests.A maximum ride time, expressed as a linear function of the direct ride time, isimposed for each user. In addition, vehicles are not allowed to be idle when carryingpassengers. A non-linear objective function combining several types of disutility isused to assess the quality of solutions. The heuristic selects users in order of earliestfeasible pick-up time and gradually inserts them into vehicle routes so as to yield


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the least possible increase of the objective function. The algorithm was tested onartificial instances involving 250 users and on a real data set with 2617 users and28 vehicles.

A commonly used technique in such problems consists of defining clusters ofusers to be served by the same vehicle, prior to the routing phase. This idea isexploited by Bodin and Sexton (1986) who construct the clusters by grouping userswho are close together in a combined space and time dimension before applyingto each cluster the single-vehicle algorithm of Sexton and Bodin (1985a,b) andmakingswapsbetweentheclusters.Resultsarepresentedontwoinstancesextractedfrom a Baltimore data base and containing approximately 85 users each. Dumaset al. (1989) later improved upon this two-phase approach by creating so-calledmini-clusters of users, i.e., groups of users to be served within the same area atapproximately the same time. These mini-clusters are then optimally combined toform feasible vehicle routes, using a column generation technique. Finally, eachvehicle route is reoptimized by means of the single-vehicle algorithm of Desrosiersetal.(1986),andaschedulingstepisexecuted.Theauthorshavesuccessfullysolvedinstances derived from real-life data taken from three Canadian cities: Montreal,Sherbrooke and Toronto. Instances with up to 200 users are easily solved, whilelarger instances require the use of a spatial and temporal decomposition technique.The mini-clustering phase was later improved by Desrosiers et al. (1991) whopresented results on a data set comprising almost 3000 users. Finally, Ioachim atal. (1995) showed there was an advantage, in terms of solution quality, to resorting

to an optimization technique to construct the clusters.A real-life problem arising in Bologna was tackled by Toth and Vigo (1996).

Users specify requests with a time window on their origin or destination. A limitproportional to direct distance is imposed on the ride time. Transportation is sup-plied by a fleet of capacitated minibuses and special cars. On occasions, taxis canbe used but since these are not the best mode of transportation for disabled people,a penalty is imposed on their use. The objective is to minimize the total cost of ser-vice. Toth and Vigo have developed a heuristic consisting of first assigning requeststo routes by means of a parallel insertion procedure, and then performing intra-

route and inter-route exchanges. Tests performed on instances involving between276 and 312 requests show significant improvements with respect to the previoushand-made solutions. Further improvements were later obtained (Toth and Vigo,1997) through the execution of a tabu thresholding post-optimization phase afterthe parallel insertion step.

Another study, by Borndrfer et al. (1997), also uses a two-phase approachin which clusters of users are first constructed and then grouped together to formfeasible vehicle routes. A cluster is defined as a maximal subtour such that thevehicle is never empty. Its two end-points correspond to the pick-up of the firstuser and the drop-off of the last user, respectively. In the first phase, a large set ofgood clusters is constructed and a set partitioning problem is then solved to selecta subset of clusters serving each user exactly once. In the second phase, feasible


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routes are enumerated by combining clusters and a second set partitioning problemis solved to select the best set of routes covering each cluster exactly once. Bothset partitioning problems are solved by a branch-and-cut algorithm. On real-lifeinstances, the algorithm cannot always be run to completion so that it must stop

prematurely with the best known solution. It was applied to instances includingbetween 859 and 1771 transportation requests per day in Berlin.

Wolfler Calvo and Colorni (2002) have devised a heuristic for a version ofthe DARP in which the number of available vehicles is fixed and windows areimposed on pick-up and drop-off times. A hierarchical objective function is used:the algorithm first attempts to service as many users as possible and then minimizesuser inconvenience expressed as the sum of waiting time and excess ride time. Theheuristic first constructs a set ofm routes and a number of subtours by solving an

assignment problem.A routing phase is then performed to insert the subtours in them routes and to resequence the vertices within the routes. Tests were carried out oninstances involving between 10 and 180 users.

The latest heuristic on the multi-vehicle static DARP is due to Cordeau andLaporte (2002). It applies tabu search to the following problem. Users specify awindow on the arrival time of their outbound trip and on the departure time of theirinbound trip, and a maximum ride time is associated with each user. It can either bethe same for all users, or computed by using a maximum deviation factor from themost direct ride time of each particular user. Capacity and maximum route length

constraints are imposed on the vehicles. The search algorithm iteratively removes atransportation request and reinserts it into another route.As is now commonly donein such contexts (Gendreau et al. 1994; Cordeau et al. 2001), intermediate infeasiblesolutions are allowed through the use of a penalized objective function. Also, theminimum duration schedule associated with each candidate solution is computed,as explained in Sect. 3.1. The algorithm was tested on randomly generated instances(24 n 144) and on six data sets (n = 200 and 295) provided by a Danishtransporter. With respect to alternative algorithms such as column generation andbranch-and-cut, tabu search can easily accommodate a large variety of constraints

and objectives, even if these are non-linear.Relatively little research on the multi-vehicle dynamic DARP is reported in the

scientific literature. One interesting case is described by Madsen et al. (1995) whohave solved a real-life problem involving services to elderly and disabled peoplein Copenhagen. Users may specify a desired pick-up or drop-off time window, butnot both. Vehicles of several types are used to provide service, not all of whichare available at all times. Requests arrive dynamically throughout the day, vehiclespeeds are variable and vehicles may become unavailable due to breakdowns. Theauthors have developed an insertion algorithm, called REBUS, based on the pro-cedure previously developed by Jaw et al. (1986). New requests are dynamicallyinserted in vehicle routes taking into account their difficulty of insertion into an ex-isting route. The algorithm was tested on a 300-customer, 24-vehicle problem. The


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Table1.

Comparisono

fseveralDARPalgorithms

Referenc

e

Type

O

bjective

Time

window

s

Other

constraints

Algorithm

Sizeofproblems

solv

ed

Psaraftis(1980)

Single-vehicle,

staticanddynamic.M

inimizeacombination

o

frouteduration,r

ide

timeandwaitingtime.

None.

Vehiclecapacity.

Maximumpositionshift.

Exact.Dynamic

programming.

n

9.

Psaraftis(1983)

Single-vehicle,

static.

M

inimizerouteduration.

Onpick

-upand

drop-off.

Vehiclecapacity.

Maximumpositionshift.

Exact.Dynamic

programming.

n

9.

Sexton(1

979),

SextonandBodin

(1985a,b

)

Single-vehicle,

static.

M

inimizeweightedsum

o

fdifferencesbetween

actualanddesired

d

rop-offtimes,and

d

ifferencesbetweenactual

andshortestpossibleride

times.

Upperb

oundson

pick-up

and

drop-offtimes.

Vehiclecapacity.

Heuristic.Iterates

betweenroutingand

schedulingphases.

7

n

20.

Desrosiersetal.

(1986)

Single-vehicle,

static.

M

inimizerouteduration.

Onpick

-upor

drop-off.

Vehiclecapacity.

Exact.Dynamic

programming.

n

40.

Jawetal.

(1986)

Multi-vehicle,

static.

M

inimizenon-linear

combinationofseveral

typesofdisutility.

Onpick

-upor

drop-off.

Vehiclecapacity.Actual

ridetimecannotexceeda

givenpercentageof

minimumridetime.

Heuristic.Insertions.

n=

250and

n=

2617.

Bodinand

Sexton(1

986)

Multi-vehicle,

static.

M

inimizeweightedsum

o

fdifferencesbetween

actualanddesired

d

rop-offtimes,and

d

ifferencesbetweenactual

andshortestpossibleride

times.

Upperb

oundson

pick-up

and

drop-offtimes.

Vehiclecapacity.

Heuristic.Iterates

betweenroutingand

schedulingphases.

n

85.


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